PHYSICAL REVIEW A 87, 033808 (2013)

Degenerate two-level system in the presence of a transverse magnetic field

L. Margalit,1 M. Rosenbluh,2 and A. D. Wilson-Gordon1

1Department of Chemistry, Bar-Ilan University, Ramat Gan 52900, Israel2The Jack and Pearl Resnick Institute for Advanced Technology, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel

(Received 2 December 2012; published 12 March 2013)

The effect of a transverse magnetic field (TMF) on the absorption spectra of degenerate two-level systemsin the D2 line of 87Rb is investigated both analytically and numerically. We compare the effect of the TMF onthe absorption of a σ polarized pump in the Hanle configuration with that of a σ− probe in the presence of aσ+ pump in the pump-probe configuration, and we show that the absorption spectra in both configurations is splitin the presence of a TMF and that the splitting is proportional to its magnitude. The population redistribution inthe ground state due to the TMF is reinforced by collisional effects, leading to increased splitting of the coherentpopulation trapping (CPT) dip in the absorption spectrum. We explain the appearance of the splitting by setting thequantization axis along the total magnetic field. Then, in the Hanle configuration, the direction of the quantizationaxis changes as the longitudinal magnetic field is scanned, so that the laser polarization also changes. This leadsto the creation of new two-photon detuned � subsystems. The evolution of the populations and coherences ofthe ground-state Zeeman sublevels is illustrated using the angular momentum probability surface. In addition tobeing split, the CPT dip in the pump-probe configuration is also shifted by the longitudinal magnetic field, sothat the effects of the longitudinal and transverse magnetic fields can be distinguished from each other.

DOI: 10.1103/PhysRevA.87.033808 PACS number(s): 42.50.Gy, 32.80.Xx, 32.70.Jz

I. INTRODUCTION

The effect of a transverse magnetic field (TMF), defined asa magnetic field perpendicular to the propagation directionof the excitation light, on the absorption and fluorescencespectra of degenerate two-level systems (DTLSs) in alkalimetals has been investigated in the past, both experimentallyand theoretically, using the Hanle configuration, in which themagnetic field is scanned through zero. Systems characterizedby coherent population trapping (CPT) [1–5], electromagneti-cally induced transparency (EIT) [6], and electromagneticallyinduced absorption (EIA) [6–10] have been studied. Thesestudies can be divided into those in which the longitudinalmagnetic field is scanned in the presence of a constant TMF[1,2,4,6,8,10] and those where the TMF is scanned in thepresence or absence of a constant longitudinal field [3,6,7,9].The influence of a TMF on the spectrum when the longitudinalmagnetic field, parallel to the light propagation direction,is scanned has been investigated in Rb for excitation withlinear, circular, and elliptical laser polarizations [1] and inCs with laser light having various elliptical polarizations [2].It was shown that the TMF can enhance or reduce the CPTamplitude depending on the orientation of the TMF withrespect to the laser polarization [1,2]. In the case of σ laserpolarization, the TMF causes a reduction in the amplitude ofthe CPT resonance and an increase in its width in a nonlinearmanner [1]. It has been shown [9] that when the TMF isscanned the Hanle resonance of a σ− probe in the presenceof a σ+ polarized pump undergoes a change of sign as thepump intensity increases. A similar effect was reported forcounterpropagating linearly polarized pump and probe beamswhen the longitudinal magnetic field is scanned [11]. In thiscase, the Hanle resonance changes sign as the angle betweenthe linear polarizations changes from 0 to 90◦.

Various studies have shown that collisional effects can havea strong influence on the spectra observed in the presenceof a TMF. For example, Renzoni et al. [7] showed that the

amplitude and width of the Hanle EIA spectrum when theTMF is scanned are strongly dependent on the ground-staterelaxation processes, and Yu et al. [4] assigned the changes inthe Hanle absorption spectrum for a circularly polarized fieldin the presence of TMF to population redistribution, due bothto the TMF and the increase in the atom-laser interaction timecaused by the buffer gas in the cell.

There are two main theoretical approaches to dealing withthe presence of a TMF in addition to a longitudinal magneticfield. In the first, the quantization axis is chosen to be along thetotal magnetic field [5,8,12], so that adding a TMF changes theorientation of the axes and thus the effective light polarization.The Zeeman shifts of the Zeeman sublevels in the Blochequations [13] are then proportional to the total magnetic field.

The second approach is to set the axis of light propagationor Bz as the quantization axis and add extra terms to the Blochequations [13] that relate to coupling between �m = ±1Zeeman sublevels due to the TMF [1,2,7,9,14]. In this ap-proach, the Zeeman shift in the Bloch equations is proportionalonly to Bz. If the Hamiltonian matrix is diagonalized, newenergy levels are obtained whose Zeeman shifts are propor-tional to the total applied magnetic field, as expected. Thus, theresults obtained from the two approaches should be identical.

To our knowledge, the effect of the TMF on the pump-probeabsorption spectra has not been investigated theoreticallyfor DTLSs. This may be due to the fact that it is simplerto solve the equations when only one laser is applied, asin the Hanle configuration. In this paper, we calculate theeffect of a TMF on the absorption of a σ polarized pumpin the Hanle configuration, and we show that it is similar tocalculating the effect of a TMF on a pump-probe configurationwith a frequency tuned σ− probe in the presence of a fixedfrequency σ+ pump in a fixed longitudinal magnetic field.For the pump-probe configuration, it is simpler to set the axisof light propagation as the quantization axis, since, for anyother arbitrary quantization axis, the initial polarizations of

033808-11050-2947/2013/87(3)/033808(7) ©2013 American Physical Society

L. MARGALIT, M. ROSENBLUH, AND A. D. WILSON-GORDON PHYSICAL REVIEW A 87, 033808 (2013)

gF =1

eF =0

1g 2g 3g

0e

σ +

1− 1+0gm

(a)

σ −

gF =1

eF =0

1− 1+0

0

π

'gm

(b)

σ −σ +

FIG. 1. (Color online) (a) The quantization axis set along the lightpropagation direction. (b) The quantization axis set along the totalmagnetic-field direction.

the pump and the probe are modified by the presence of theTMF and become combinations of σ+, σ−, and π , so that bothfields interact with the same transitions, making the problemdifficult to solve. Thus, in our calculations we set the axis oflight propagation to be the quantization axis.

In the case of a pump-probe configuration, in which eachfield interacts with a different hyperfine transition, as inthe three-level � system, the CPT spectrum can be easilycalculated using an arbitrary quantization axis. In this case, thecombination of σ+, σ−, and π polarizations in the presence ofthe TMF leads to the creation of multiple frequency displaced� systems and therefore to new resonances in the CPT spectra[12,15]. In [12,15], the quantization axis was chosen to bealong the total magnetic field, and we have confirmed theirresults using the calculational model reported here, in whichthe quantization axis is set to be along the light propagationdirection.

Here, we show both numerically and analytically thatthe absorption spectra in both the Hanle and pump-probeconfigurations for the Fg = 1 → Fe = 0 transition (see Fig. 1)are split in the presence of a TMF and that the splitting isproportional to the magnitude of the TMF. We note that thesplitting is more marked in the presence of collisions thatreinforce the effect of population redistribution due to themixing of the Zeeman sublevels by the TMF. As in the case ofthe degenerate three-level � system [12,15], the splitting canbe explained as being due to the creation of new � subsystemsformed by the Zeeman sublevels.

In addition to being split, the CPT line in the pump-probeconfiguration is shifted by the longitudinal magnetic field.Thus, we find that the effects of longitudinal and transversemagnetic fields can be distinguished from each other.

II. THE OPTICAL BLOCH EQUATIONS

Choosing the propagation axis z as the quantization axis, wewrite the Bloch equations for a system consisting of a groundhyperfine state Fg composed of 2Fg + 1 Zeeman sublevels gi

and an excited hyperfine state Fe composed of 2Fe + 1 Zeemansublevels ej , interacting with one or more electromagneticfields in the presence of a constant longitudinal magneticfield. We use the equations for the time evolution of theZeeman sublevels as formulated by Renzoni et al. [16], withthe addition of decay from the ground and excited states toa reservoir [17], and collisions between atoms in the Zeemansublevels of the ground states [18].

The optical Bloch equations have the following form:

ρ̇ei ej= −(

iωeiej+ �

)ρeiej

+ (i/h̄)∑gk

(ρeigk

Vgkej− Veigk

ρgkej

)− γ

(ρeiej

− ρeqeiei

)δeiej

, (1)

ρ̇eigj= −(

iωeigj+ �′

eigj

)ρeigj

+ (i/h̄)

(∑ek

ρeiekVekgj

−∑gk

Veigkρgkgj

), (2)

ρ̇gigi= (i/h̄)

∑ek

(ρgiek

Vekgi− Vgiek

ρekgi

)+ (

ρ̇gigi

)SE

− γ(ρgigi

− ρeqgigi

)+

∑gk,k �=i

�gkgiρgkgk

− �giρgigi

, (3)

ρ̇gigj= −(

iωgigj+ �′

gigj

)ρgigj

+ (i/h̄)

×∑ek

(ρgiek

Vekgj− Vgiek

ρekgj

) + (ρ̇gigj

)SE, (4)

where

(ρ̇gigj

)SE = (2Fe + 1)�Fe→Fg

∑q=−1,0,1

Fe∑me,m′

e=−Fe

(−1)−me−m′e

×(

Fg 1 Fe

−mgiq me

)ρme m′

e

(Fe 1 Fg

−m′e q mgj

),

(5)

with

�Fe→Fg= (2Fg + 1)(2Je + 1)

{Fe 1 Fg

Jg I Je

}2

� ≡ b�.

(6)

In Eqs. (1)–(4), � is the total spontaneous emission rate fromeach Feme sublevel, whereas �Fe→Fg

is the decay rate from Fe

to one of the Fg states. When b = 1 the system is closed,whereas when b < 1 the system is open. �gi

is the totalcollisional decay rate from sublevel gi, �gigj

is the rate oftransfer from sublevel gi → gj , and γ is the rate of decay dueto time of flight through the laser beams. The dephasing rates

033808-2

DEGENERATE TWO-LEVEL SYSTEM IN THE PRESENCE . . . PHYSICAL REVIEW A 87, 033808 (2013)

of the excited- to ground-state coherences are given by �′eigj

=γ + 1

2 (� + �gj) + �∗, where �∗ is the rate of phase-changing

collisions. The dephasing rates of the ground-state coherencesare given by �′

gigj= γ + 1

2 (�gi+ �gj

) + �∗gigj

, where �∗gigj

is the rate of phase-changing collisions. The Zeeman splitting�EmF

= μBgF BzmF of the ground and excited levels due toan applied Bz magnetic field, where μB is the Bohr magnetonand gF is the gyromagnetic factor of the ground or excitedstate, is included in the frequency separation between the levelsai and bj , given by ωaibj

= (Eai− Ebj

)/h̄, with a,b = (g,e).ρ

eqaiai

with a = (g,e) is the equilibrium population of state ai,

in the absence of any electromagnetic fields.In this paper, we solve the general Bloch equations in

the steady state for two cases: (1) a pump of frequency ω1

with σ polarization and a scanning longitudinal magnetic fieldBz in the Hanle configuration and (2) a pump of frequencyω1 with σ+ polarization and a scanning probe of frequencyω2 with σ− polarization in the pump-probe configuration. Inboth cases, we consider the effect of an additional transversemagnetic field, for instance Bx . Thus, we include the followingadditional terms [7] in the Bloch equations:

ρ̇eiej|Bx

= −iμBBx

2h̄ge

{c+eiρei+1ej

+ c−eiρei−1ej

− c+ej

ρeiej+1 − c−ej

ρeiej−1

}, (7)

ρ̇eigj|Bx

= −iμBBx

2h̄

{ge

[c+eiρei+1gj

+ c−eiρei−1gj

]− gg

[c+gj

ρeigj+1 + c−gj

ρeigj−1

]}, (8)

ρ̇gigj|Bx

= −iμBBx

2h̄gg

{c+giρgi+1gj

+ c−giρgi−1gj

− c+gj

ρgigj+1 − c−gj

ρgigj−1

}, (9)

where c±FmF

≡ √(F ∓ mF )(F ± mF + 1).

The TMF couples the mF Zeeman sublevels in incrementsof �mF ± 1 and causes redistribution of the population amongthe Zeeman sublevels, as do the collisional and time-of-flightterms in Eqs. (1) and (3).

From the solution of the Bloch equations, we calculate theabsorption of the system. The pump and probe absorptionα(ω1,2) is given by [19,20]

α(ω1,2) = 4πω0N

h̄c

∑eigj

∣∣μeigj

∣∣2

Veigj(ω1,2)

Im[ρeigj(ω1,2)], (10)

where N is the atomic density and ω0 is the transition frequencyin the absence of a magnetic field.

III. THE HANLE CONFIGURATION

A. Analytical solutions of Bloch equations

For the Hanle configuration, we solve the Bloch equationsanalytically for the Fg = 1 → Fe = 0 system in the presenceof a scanning longitudinal magnetic field Bz and a constantTMF Bx , for a linearly polarized pump whose σ polarizationis parallel to the TMF. The pump is resonant with the |Fg =1,mg = 0〉 → |Fe = 0,me = 0〉 transition [see Fig. 1(a)]. Asthe longitudinal magnetic field is scanned, the detuningschange such that �eg1 = −�eg3 . It can be shown that, in the

absence of the TMF, ρg1g1 = ρg3g3 , ρee ≈ 0, and ρeg1 = −ρ∗eg3

[21], and also that ρg1g2 = −ρg2g3 and ρeg2 = 0 [22]. We canshow numerically and analytically that these relations still holdeven in the presence of the small TMFs considered here, so thatthe sixteen Bloch equations can be reduced to three equations:

ρ̇eg1 = (i�1 − �′)ρeg1 + iV(ρg1g1 − ρg3g1

), (11)

ρ̇g3g1 = (i2ω − γ ′)ρg3g1 − 2iVρeg1 − 2iAρg2g1 , (12)

ρ̇g2g1 = (iω − γ ′)ρg2g1 + iVρg2e + iA(1 − 3ρg1g1 − ρg3g1

),

(13)

where ω = ωg1g2 = −μBBzgg/h̄, �1 = ω1 − ωeg1 = ωg1g2 =ω, �′ = �′

eigj, γ ′ = �′

gigj, and A = μBBxgg/

√2h̄. The steady-

state solution of ρeg1 is given by

ρeg1 = − iVρg1g1

(i�1 − �′)D−1 − 2iA2V

(1 − 3ρg1g1

)(i�1 − �′)d

D−1, (14)

where

D = 1 + 2V 2d + 4A2V 2

(i�1 − �′)(−2iω + γ ′)d, (15)

with

d = [(−iω + γ ′)(−2iω + γ ′) + 2A2]. (16)

When we consider the denominators in Eq. (14), we seefrom the expression (i� − �′) = (iω − �′) that the pumpabsorption has a maximum when ω = 0 where Bz = 0, andwe see from the expression for d [see Eq. (16)] that the pumpabsorption has minima at ω = ±A where Bz = ±Bx/

√2.

We note that the minima occur at the same values of Bz

where the population is equally distributed among the Zeemanground-state sublevels, so that the second term in Eq. (14)becomes zero.

B. Results and discussion of the Hanle effect

In Fig. 2, we show numerical results for the absorptionas a function of longitudinal field for a σ polarized resonantlaser exciting a realistic closed transition, Fg = 1 → Fe = 0of the 87Rb D2 line (similar results were obtained for theFg = 1 → Fe = 1 and Fg = 2 → Fe = 1 open transitions),in the absence and presence of a TMF. In the absence of thetransverse field (solid line), the sublevels of the lower levelare all degenerate at Bz = 0, leading to two-photon resonance,and CPT is observed. This is reflected in a narrow dip in thebroad absorption spectrum at Bz = 0.

In the presence of a constant TMF (dashed line), thespectrum of the pump absorption is modified: the narrowCPT dip widens, its amplitude decreases, and it splits into twosymmetrically displaced dips at Bz = ±Bx/

√2, as predicted

analytically. Thus, the distance between the dips increaseslinearly with Bx (see the inset in Fig. 2). This phenomenoncan effectively be implemented in magnetometry, as it allowsmeasurement of the value of the TMF by determining theseparation between the dips. Similar results have been shownfor the Fg = 2 → Fe = 1 transition in the 87Rb D1 line foran elliptically polarized pump [1]. The authors suggest thatthis effect is due to the creation of high-order ground-state

033808-3

L. MARGALIT, M. ROSENBLUH, AND A. D. WILSON-GORDON PHYSICAL REVIEW A 87, 033808 (2013)

−0.1 −0.05 0 0.05 0.10.2

0.4

0.6

0.8

1

1.2

Bz (G)

Abs

orpt

ion

(cm

−1 )

0 0.25 0.50

5

10

15

Bx (G)

Spl

ittin

g (M

Hz)

Bx=0

Bx=0.01G

FIG. 2. (Color online) Hanle effect in Fg = 1 → Fe = 0 of the87Rb D2 line. The scanning magnetic field is Bz, and the additionalconstant transverse magnetic field (Bx = 0.01 G) causes the narrowCPT dip to widen and split. The parameters used in the calculationare � = 6π × 106 s−1, � = 2π × 6.0666 MHz, γ = 10−5�, �∗ =100�, �∗

gigj= 0, and �gigj

= 10−5�. When �gigj= 0, the contrast

of the splitting is smaller, as the collisional population redistributionreinforces the redistribution caused by the TMF. Inset: The splittingas a function of Bx , calculated numerically with the full Blochequations.

coherences [23], �m = 4 in their case, so that it should notbe expected to occur for the Fg = 1 → Fe = 0 transition.However, we show that the splitting does indeed occur forthe Fg = 1 → Fe = 0 transition. This splitting was not seenin the earlier work of [1] due to the absence of collisionsin their experimental system. These collisions reinforce theredistribution of the population in the Zeeman sublevels dueto the TMF.

Splitting of an EIT dip by a TMF has also been observedfor an N system in Rb atoms [24], and splitting of a CPT diphas been observed for a tilted magnetic field in a Rb atomicmagnetometer that uses polarization modulation to eliminate“dead zones,” that is, orientations of the magnetic field wherethe magnetometer loses its sensitivity [25].

The redistribution of the population by the combined effectsof the TMF and the collisions is illustrated in Fig. 3, in whichwe see that the population is trapped in the |Fg = 1,mg = ±1〉Zeeman sublevels. In Fig. 4, we see that the TMF has createdthe ground-state coherences ρg1g2 and ρg2g3 that were zero in itsabsence. In addition, the degree of coherence of the subsystemsSg1g2 = Sg2g3 , where Sgigj

= (ρgigj)2/ρgigi

ρgj gj, is maximum

close to the points of the minima in the absorption spectra,whereas Sg1g3 remains maximum at Bz = 0.

In order to explain physically the origin of the splitting,we will consider the problem in the basis set in which thedirection of the total magnetic field is the quantization axisof the system [Fig. 1(b)]. We will refer to these ground-stateZeeman sublevels as m′

g to distinguish them from the mg basisdiscussed above, where the quantization is in the z direction[Fig. 1(a)]. In the absence of a TMF, the quantization axis isaligned along the z axis. The light propagates in the z direction,so that the laser polarization can be considered as purely σ .However, in the presence of a constant TMF, the direction ofthe quantization axis changes as Bz is scanned, so that the laser

−0.1 −0.05 0 0.05 0.10.1

0.2

0.3

0.4

0.5

Bz (G)

PO

PU

LAT

ION

of F

g=1,

mF=

±1

−0.1 −0.05 0 0.05 0.10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Bz (G)

g=1,

mF=

0

(b)(a)

PO

PU

LAT

ION

of F

FIG. 3. (Color online) The populations in the (a) |Fg = 1,mg =±1〉 and (b) |Fg = 1,mg = 0〉 Zeeman sublevels for the sameparameters as in Fig. 2, in the absence (solid line) and presence(dashed line) of TMF Bx = 0.01 G.

polarization also changes. Thus, the polarization is no longerpurely σ but is a combination of σ and π .

At Bz = 0, in the presence of a TMF, the laser is purelyπ polarized, leading to absorption in the |Fg = 1,mg = 0〉 →|Fe = 0,me = 0〉 transition. This is reflected in the absorptionpeak that appears when Bz = 0 (the dashed line in Fig. 2).

When Bz �= 0, σ polarization is added to the π po-larization, leading to two additional two-photon detuned� subsystems: |Fg = 1,m′

g = −1〉 ↔ |Fe = 0,m′e = 0〉 ↔

|Fg = 0,m′g = 0〉 and |Fg = 1,m′

g = 1〉 ↔ |Fe = 0,m′e =

0〉 ↔ |Fg = 0,m′g = 0〉. As Bz increases, the transitions that

interact with the σ polarization become more detuned due tothe increase in the Zeeman splitting of the m′

g sublevels. On theother hand, the relative intensity of the σ polarized componentof the light increases. When the σ polarized component of

−0.1 −0.05 0 0.05 0.10

0.2

0.4

0.6

0.8

Deg

ree

of c

oher

ence

Bz (G)

−0.1 −0.05 0 0.05 0.1

0.9

1

1.1

Abs

orpt

ion

(cm

−1 )

Sg

1g

3

Sg

1g

2

absorption

FIG. 4. (Color online) Degree of coherence calculated numeri-cally from the full Bloch equations for the same parameters as inFig. 2, in the presence of a TMF (Bx = 0.01G). Sg1g2 is maximumclose to the points of the minima in the absorption spectra, whereasSg1g3 has a higher maximum at Bz = 0.

033808-4

DEGENERATE TWO-LEVEL SYSTEM IN THE PRESENCE . . . PHYSICAL REVIEW A 87, 033808 (2013)

the light is sufficiently strong, the two competing effectslead to a region where two-photon detuned CPT occurs. Weverified this description by considering a simple � systemin which the field interacting with one of the legs graduallybecomes stronger and more detuned while the other fieldremains at resonance but gradually becomes weaker. As Bx in-creases, the absorption spectrum becomes broader and weakeruntil the splitting becomes indiscernible. In this picture[Fig. 1(b)], the system is similar to the tripod system in whichthe Fg = 1 → Fe = 0 transition interacts with a σ polarizedpump and a π polarized probe [22,26]. The same phenomenonoccurs for the Fg = 2 → Fe = 1 transition where the TMFleads to the formation of a multitripod system [27,28].

IV. PUMP-PROBE CONFIGURATION

A. Analytical solutions of Bloch equations

As in the case of the Hanle configuration, we analyticallysolve the Bloch equations for the pump-probe configurationfor the Fg = 1 → Fe = 0 system for a σ+ polarized pumpresonant with the |Fg = 1,mg = 0〉 → |Fe = 0,me = 0〉 tran-sition and a scanning σ− polarized probe in the presence ofconstant longitudinal and transverse magnetic fields.

Based on the assumptions given in Sec. III A, the 16 Blochequations can be reduced to the following three equations:

ρ̇eg3 = [i(�2 − δ) − �′]ρeg3 + iV(ρg1g3 + ρg3g3

), (17)

ρ̇g1g3= [i(�2 − 2δ) − γ ′]ρg1g3+ 2iVρeg3− iA(ρg2g3− ρg1g2

),

(18)

ρ̇g1g2 = [i(�2/2 − δ) − γ ′]ρg1g2 + iVρeg2

− iA(1 − 3ρg3g3 − ρg1g3

), (19)

where �2 = ω2 − ωeg2 and δ = μBBzgg/h̄. Note that in theHanle configuration ω1 is fixed and �1 changes with Bz,whereas in the pump-probe configuration ω1 and Bz are fixedand �2 varies with ω2. The steady-state solution of ρeg3 isgiven by

ρeg3 = − iVρg3g3

i(�2 − δ) − �′ D−1 − 2iA2V

(1 − 3ρg3g3

)[i(�2 − δ) − �′]d

D−1,

(20)

where

D ={[i(�2 − δ) − �′][i(�2 − 2δ) − γ ′] + 2V 2}d − 4A2V 2

[i(�2 − δ) − �′][i(�2 − 2δ) − γ ′]d,

(21)

with

d = {[(i�2/2 − δ) − γ ′][i(�2 − 2δ) − γ ′] + 2A2}. (22)

When we consider the denominators in Eq. (20), we see fromthe expression i(�2 − δ) − �′ that the pump absorption hasa maximum when �2 = 0, and we see from the expressionfor d [see Eq. (22)] that the pump absorption has minima at�2 = 2δ ± 2A. It is interesting to note that Bx and Bz haveindependent effects on the positions of the minima.

−0.5 0 0.5 1 1.5

0.1

0.15

0.2

0.25

0.3

0.35

Δ2 (MHz)

Pro

be a

bsor

ptio

n (c

m−

1 )

b

c d

a

FIG. 5. (Color online) Probe absorption in the realistic systemFg = 1 → Fe = 0 of the 87Rb D2 line for different values of thelongitudinal and transverse magnetic fields. (a) Dotted line: Bz = 0and Bx = 0. (b) Dot-dashed line: Bz = 0.1 G and Bx = 0. (c) Dashedline: Bz = 0 and Bx = 0.01 G. (d) Continuous line: Bz = 0.1 G andBx = 0.01 G. The parameters used in the calculation are � = 6π ×106 s−1, � = 2π × 6.067 MHz, γ = 10−5�, �∗ = 100�, �∗

gigj= 0,

and �gigj= 10−5�.

B. Results and discussion of the pump-probe configuration

In Fig. 5, we show numerical results for the probeabsorption in the realistic system Fg = 1 → Fe = 0 of the87Rb D2 line for a σ+ polarized pump and a scanningσ− polarized probe for different values of the longitudinaland transverse magnetic fields (similar results are obtained forthe Fg = 1 → Fe = 1 and Fg = 2 → Fe = 1 systems). In theabsence of any magnetic field (case a), the CPT dip is centeredat � = 0. When only a longitudinal magnetic field is applied(case b), Zeeman splitting of the ground hyperfine state occursand the CPT dip is shifted to �2 = 2δ = 2μBggBz. When onlya transverse magnetic field exists (case c), the CPT dip widensand is split into two with reduced amplitude. When both ofthe fields are present (case d), the CPT dip is both shifted andsplit. Generally, we can say that Bz determines the generalshift of the CPT dip, while Bx causes the splitting of the dip.As predicted analytically, the longitudinal and the transversemagnetic fields lead to two different independently measurablephenomena. Thus, this effect can be applied successfully invector magnetometry.

If we set the quantization axis to be along the total magneticfield, in the pump-probe configuration the direction of thequantization axis is constant and determined by the fixedlongitudinal and transverse magnetic fields, unlike the Hanleconfiguration, in which the direction of the quantization axisvaries as a function of the scanned field. The longitudinalmagnetic field shifts the Zeeman sublevels in energy but doesnot change the laser polarization, which remains purely σ+for the pump and σ− for the probe. The TMF, however, inaddition to the shift in energy of the Zeeman sublevels, changesthe laser polarization from pure σ+ and σ− to a combinationof σ+, σ−, and π polarizations, thereby creating two new� subsystems which lead to the two dips in the absorptionspectrum.

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−0.1 0 0.1

0.9

1.1

Bz (G)

Abs

orpt

ion

(cm

−1 )

b

c

d

(a) (b)

y

z

x

(c)

y

z

x

(d)

y

z

x

FIG. 6. (Color online) The AMPS at various values of Bz in theHanle type spectrum of Fig. 2. (a) Pump absorption in the Hanleconfiguration in the presence of Bx = 0.01G (the same spectrum asin Fig. 2), (b) far from resonance (|Bz| > 0), (c) near the minima atBz = ±Bx/

√2, (d) at resonance (Bz = 0).

V. ANGULAR MOMENTUM PROBABILITYSURFACE (AMPS)

The ground-state coherence created in the presence ofthe TMF can be illustrated using the angular momentumprobability surface (AMPS) [29], whose radius in a givendirection is determined by the probability of measuring themaximum possible angular momentum projection in thatdirection.

The AMPS evolution of a CPT process in the pump-probeconfiguration for a fixed pump as a function of the probedetuning in the absence of a magnetic field (which is equivalentto scanning Bz in the Hanle configuration) has been illustratedin a movie by Rochester and Budker [30]. There, we seethat when the probe is far from resonance there is almostno coherence between the Zeeman sublevels and due tooptical pumping most of the population is concentrated in the|Fg = 1,mg = 0〉 sublevel, causing the AMPS to resemble a“doughnut” aligned along the z axis. As the probe approachesresonance, the ground-state coherence ρg1g3 increases so thatthe probability distribution now resembles a “peanut” alongthe x axis.

In Fig. 6, we show the AMPS for various values of Bz

in the Hanle type spectrum of Fig. 2 [31]. Here, the AMPSevolution of the CPT process is affected by the ground-statecoherences between neighboring sublevels that are created bythe TMF. Far from resonance (|Bz| > 0), there is no significantchange from the case of the CPT process in the absenceof TMF, so that the AMPS has the same doughnut shapealigned along the z axis. Near the minima at Bz = ±Bx/

√2,

the AMPS has a doughnut shape tilted with respect to the

z axis. At this point, the population is equally distributedamong the Zeeman sublevels, and the amplitudes of thecoherences between the various sublevels are approximatelyequal (ρg1g2 ≈ −ρg2g3 ≈ ρg1g3 ≈ −1/3). If all the coherenceswere zero, the AMPS would be spherical (the unpolarizedstate), and, if only the coherence between the extreme levelswere nonzero, the AMPS would be peanut shaped along thex axis, as in the case of CPT. On the other hand, if onlythe coherences between neighboring states existed, the peanutwould be tilted with respect to the z axis. When all thecoherences are present, the peanut turns into a doughnut withthe same tilt. At resonance (Bz = 0), the population is in theextreme levels and the only nonzero coherence is betweenthese levels. This results in a doughnut-shaped AMPS alignedalong the x axis.

VI. CONCLUSIONS

The effect of a TMF on the absorption spectra of degeneratetwo-level systems (DTLSs) of the 87Rb D2 line has beenanalyzed both analytically and numerically. We compared theeffect of the TMF on the absorption of a σ polarized pump inthe Hanle configuration with that of a σ− probe in the presenceof a σ+ pump in the pump-probe configuration, and we showedthat the absorption spectra in both the Hanle and pump-probeconfigurations are split in the presence of a TMF and that thesplitting is proportional to the magnitude of the TMF. We notethat the splitting is more marked in the presence of collisionsthat reinforce the effect of population redistribution due to theTMF.

In our calculations, we set the axis of light propagationto be the quantization axis. This gives the same results assetting the quantization axis in any other arbitrary direction.The change in the effective light polarization in the case inwhich the quantization axis is set along the total magneticfield leads to the creation of new two-photon detuned �

subsystems formed by the Zeeman sublevels. There are twocompeting effects that lead to a region where two-photondetuned CPT occurs. In addition to being split, the CPT dipin the pump-probe configuration is shifted by the longitudinalmagnetic field. Thus, we find that the effects of longitudinaland transverse magnetic fields can be distinguished from eachother; that is, the value and sign of Bz and the value of Bx canbe determined separately. In the future, we will investigatethe effect of adding By to the already existing magneticfields, in order to explore the possibility of achieving vectormagnetometry with all components of the magnetic field.

ACKNOWLEDGMENTS

We are grateful to D. Budker and F. Renzoni for stimulatingdiscussions.

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